WEBVTT

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[Music]

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My name is Jim Lewis.

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I am professor of mathematics
here at the University

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of Nebraska-Lincoln and also
the director of the Center

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for Science, Math &
Computer Education.

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Over the past year, year
and a half, I have worked

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with the fractions panel

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to create a fractions practice
guide for Doing What Works.

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When we encounter a situation
we want to understand

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or a problem we want to solve,

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sometimes multiplicative
relationships matter,

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and that's what people
mean when they talk

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about thinking proportionally.

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You have a photograph, you want
the photograph blown up to be

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on a poster that's going
to advertise something.

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You know the photograph is
about three inches tall,

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and when you blow
it up it's going

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to be about 12 inches tall.

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It's shocking how many people
don't realize the width changes

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as well.

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We need to think about
multiplicative relationships

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and the ratio between quantities
when we are changing like this.

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In order for students to be
successful and working problems

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that call for proportional
reasoning,

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we have got to take them on
a journey from where they are

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when they start to where
we want them to get.

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To go on a journey so they
understand the concepts

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involved, we want them to see
this proportional relationship

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with much simpler numbers.

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We want them to see the build-up
strategy and a ratio table.

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But then we want them to
see their first problem

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that is what I called
"going by way of one" or a unit ratio strategy.

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They see the context as the
difficulty or the complexity

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of the numbers involved and the
question being asked continues

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to increase.

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And so they see the ratios
that they want to set

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up with much simpler problems
and much simpler questions.

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So as the problem becomes
more complex, they use units

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of measure, the context,
and past experience

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with similar problems to
set up a ratio that is going

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to successfully lead
them to the answer.

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A teacher will craft the journey

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from a problem that's
fairly easy to think

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about to one that's
more involved.

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And then the teacher will begin
to show the fractions that are

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in the problem, the
two-fifths representing the

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two-to-five ratio.

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The ratio table is a
technique that allows one

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to see several values that
often represent the same ratio.

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And how corresponding fractions
are equal can lay a foundation

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for students to understand
context and to move

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in the direction of setting up
ratios that become proportions.

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When we are saying a first ratio
needs to equal a second ratio,

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then that's a proportion
that we might want to solve.

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Students know that they write
fractions and compare fractions,

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but sometimes they don't quite
know how they are supposed

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to compare them.

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Sometimes with very
concrete number problems,

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we get into comparison and
students don't understand what

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to do with the comparison.

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Students sometimes don't
recognize the problem

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as the same mathematically
if you change the context,

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or they don't recognize it
as the same mathematically

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if you made the numbers involved
slightly more complicated

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than they are used to seeing.

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If they have learned that
they should set a proportion

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with the missing answer-the
problem they want to solve

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as a variable-they
sometimes set up a proportion,

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but they setup the
wrong proportion

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because the fractions they
write don't really mean anything

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to them.

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It would help if teachers
start emphasizing units.

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Sometimes I see students

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who choose the location simply
wanting the variable to be

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in the numerator rather than
matching up the context.

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If teachers emphasize the
unit-flour to water or water

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to flour equals water to
flour-then the student knows

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that they have to
have the numbers

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and the variable match
with those units.

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This is typical of both the kind

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of problem students will run
into-they know they want a

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proportion, but they don't know

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which proportion-and how
teachers can work to minimize

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that kind of difficulty.

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The ratio table helps.

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Using units in the fractions
that you've set up helps.

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As the problems get more
difficult, we've got to set

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up a proportion and we've got
to have a technique for solving

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that proportion, because
we can't just think

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about the numbers in our head
and reason our way to an answer.

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But getting the right
proportion is the central piece

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of the problem.

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What you do after that
point is just technique.

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People refer to it
as cross-multiplying.

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It's fairly straightforward,
but if you're working

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with the wrong ratios,
it's not going

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to give you the correct answer.

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It's surprising how often
I've worked with teachers

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and asked them, "Why does
cross-multiplying work?"

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What I want them to
understand is it really is

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about answering a question,
"When are two fractions equal?"

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or "Which fraction is bigger?"

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If I gave you the fractions
48 over 64 and 36 over 48,

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a sixth-grade student
and perhaps many adults

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and some teachers do not quickly
and easily answer that question,

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"What is the ratio, or
what is the fraction

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that they are looking at?"

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But they do learn a technique.

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If you have two fractions and if
they have the same denominator,

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then they are equal only if
they have the same numerator.

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So the goal is to move from
the fractions you're working

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with to a pair of
equivalent fractions

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that have the same denominator,

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and now you'll compare
numerators.

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That's all cross-multiplication
is.

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Those are the two
numerators you get

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when you get a common
denominator

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for the proportion
that you set up.

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As teachers are working with
students, and they are beginning

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to work some problems
correctly, the teacher wants

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to equip the student with
enough strategies-essentially,

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enough tools in their toolbox-so
that they will be able

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to solve a wide variety
of problems.

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[Music]